Am I supposed to eat back calories burned?

Yes, if you're following MFP's plan. Some people feel, though, that MFP overestimates calories burned and therefore eat a fraction of them back. The most common fraction is 1/2, from what I have seen people say.

I try to burn a minimum of 300 calories a day, and sometimes a lot more! I keep track of calories burned using my Ft4 HRM (pretty accurate, I entered my age/height/heart rate for an activity and the duration into "calories burned" equations on fitness websites and very similar results for calories burned) - so I know that the calories are not too over or underestimated. My calorie goal is 1,200. I tend to eat back 100 to 200 calories of the exercise calories. I don't eat them all back just in case I underestimated some of the calories of food (like if I actually had 1.2 servings instead of 1 of something). I do intentionally eat back at least 100 calories just in case I under-estimated my calories - I like to make sure I get enough nutrition and calories to build muscle. I may eat all of them back if it is a special day or if I am feeling weak (your body needs a certain amount of calories to get stronger/build muscle). Listen to your body. Eat to strengthen, hold off on eating all of them if you feel good and you are seeing improvement in your strength despite the calorie shortage.